Step |
Hyp |
Ref |
Expression |
1 |
|
eqg0subg.0 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
2 |
|
eqg0subg.s |
⊢ 𝑆 = { 0 } |
3 |
|
eqg0subg.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
4 |
|
eqg0subg.r |
⊢ 𝑅 = ( 𝐺 ~QG 𝑆 ) |
5 |
1
|
0subg |
⊢ ( 𝐺 ∈ Grp → { 0 } ∈ ( SubGrp ‘ 𝐺 ) ) |
6 |
3
|
subgss |
⊢ ( { 0 } ∈ ( SubGrp ‘ 𝐺 ) → { 0 } ⊆ 𝐵 ) |
7 |
5 6
|
syl |
⊢ ( 𝐺 ∈ Grp → { 0 } ⊆ 𝐵 ) |
8 |
2 7
|
eqsstrid |
⊢ ( 𝐺 ∈ Grp → 𝑆 ⊆ 𝐵 ) |
9 |
|
eqid |
⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 ) |
10 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
11 |
3 9 10 4
|
eqgfval |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ) → 𝑅 = { 〈 𝑥 , 𝑦 〉 ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐵 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ) } ) |
12 |
8 11
|
mpdan |
⊢ ( 𝐺 ∈ Grp → 𝑅 = { 〈 𝑥 , 𝑦 〉 ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐵 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ) } ) |
13 |
|
opabresid |
⊢ ( I ↾ 𝐵 ) = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥 ) } |
14 |
|
simpl |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥 ) → 𝑥 ∈ 𝐵 ) |
15 |
|
eleq1w |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵 ) ) |
16 |
15
|
equcoms |
⊢ ( 𝑦 = 𝑥 → ( 𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵 ) ) |
17 |
16
|
biimpac |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥 ) → 𝑦 ∈ 𝐵 ) |
18 |
|
simpr |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥 ) → 𝑦 = 𝑥 ) |
19 |
14 17 18
|
jca31 |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥 ) → ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑦 = 𝑥 ) ) |
20 |
|
simpl |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → 𝑥 ∈ 𝐵 ) |
21 |
20
|
anim1i |
⊢ ( ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑦 = 𝑥 ) → ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥 ) ) |
22 |
21
|
a1i |
⊢ ( 𝐺 ∈ Grp → ( ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑦 = 𝑥 ) → ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥 ) ) ) |
23 |
19 22
|
impbid2 |
⊢ ( 𝐺 ∈ Grp → ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥 ) ↔ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑦 = 𝑥 ) ) ) |
24 |
|
simpl |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝐺 ∈ Grp ) |
25 |
|
simpr |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → 𝑦 ∈ 𝐵 ) |
26 |
25
|
adantl |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑦 ∈ 𝐵 ) |
27 |
20
|
adantl |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑥 ∈ 𝐵 ) |
28 |
3 9 24 26 27
|
grpinv11 |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) = ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ↔ 𝑦 = 𝑥 ) ) |
29 |
3 9
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝐵 ) |
30 |
29
|
adantrr |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝐵 ) |
31 |
3 10 1 9
|
grpinvid2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝐵 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) = ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ↔ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) = 0 ) ) |
32 |
24 26 30 31
|
syl3anc |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) = ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ↔ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) = 0 ) ) |
33 |
28 32
|
bitr3d |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑦 = 𝑥 ↔ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) = 0 ) ) |
34 |
33
|
pm5.32da |
⊢ ( 𝐺 ∈ Grp → ( ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑦 = 𝑥 ) ↔ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) = 0 ) ) ) |
35 |
|
vex |
⊢ 𝑥 ∈ V |
36 |
|
vex |
⊢ 𝑦 ∈ V |
37 |
35 36
|
prss |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ↔ { 𝑥 , 𝑦 } ⊆ 𝐵 ) |
38 |
37
|
a1i |
⊢ ( 𝐺 ∈ Grp → ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ↔ { 𝑥 , 𝑦 } ⊆ 𝐵 ) ) |
39 |
2
|
eleq2i |
⊢ ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ↔ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ { 0 } ) |
40 |
|
ovex |
⊢ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ V |
41 |
40
|
elsn |
⊢ ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ { 0 } ↔ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) = 0 ) |
42 |
39 41
|
bitr2i |
⊢ ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) = 0 ↔ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ) |
43 |
42
|
a1i |
⊢ ( 𝐺 ∈ Grp → ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) = 0 ↔ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ) ) |
44 |
38 43
|
anbi12d |
⊢ ( 𝐺 ∈ Grp → ( ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) = 0 ) ↔ ( { 𝑥 , 𝑦 } ⊆ 𝐵 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ) ) ) |
45 |
23 34 44
|
3bitrd |
⊢ ( 𝐺 ∈ Grp → ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥 ) ↔ ( { 𝑥 , 𝑦 } ⊆ 𝐵 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ) ) ) |
46 |
45
|
opabbidv |
⊢ ( 𝐺 ∈ Grp → { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥 ) } = { 〈 𝑥 , 𝑦 〉 ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐵 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ) } ) |
47 |
13 46
|
eqtr2id |
⊢ ( 𝐺 ∈ Grp → { 〈 𝑥 , 𝑦 〉 ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐵 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ) } = ( I ↾ 𝐵 ) ) |
48 |
12 47
|
eqtrd |
⊢ ( 𝐺 ∈ Grp → 𝑅 = ( I ↾ 𝐵 ) ) |